however, here the regularization parameters \(\sigma _\textup {v}\)  and
\(\sigma _\textup {w}\)  have a probabilistic meaning: \(\sigma _\textup {v}^2\)  models
the uncertainty caused by noise in the training data, while \(\sigma _\textup {w}^2\) 
is the variance of the prior distribution of the regression coefficients. These values
are optimized using the evidence approximation [1]}, [2]} and
are generally invoked to prevent overfitting. Moreover, the
probabilistic framework of BLR is advantageous for the on-the-fly training.
In fact, under the assumption that the prior distribution of the target data (that is,
the ab initio energies, forces and stress tensors) follows a multivariate
Gaussian distribution, from Bayesian inference it derives that the posterior
distribution of the predicted data is a Gaussian distribution too. The BLR
framework then allows us to define a `prediction variance' that measures the
uncertainty of the predictions. For any given configuration,
this is estimated as the variance of the posterior distribution:
\(\sigma ^2 = \sigma _\textup {v}^2 \,\mathbf {I} +\sigma _\textup {v}^2 \,\varphi \,\Big ( \Phi ^T \Phi +\sigma _\textup {v}^2/ \sigma _\textup {w}^2 \,\mathbf {I} \Big )^{-1} \varphi ^T,\) 
